A Geometric Criterion for the Weaker Principle of Spatial Averaging

Commun. Korean Math. Soc. 1999 Vol. 14, No. 2, 337-352

HyukJin Kwean Korea University

Abstract : In this paper we find a geometric condition for the weaker principle of spatial averaging (PSA) for a class of polyhedral domains. Let $\Omega_n$ be a polyhedron in $R^3,\ n\le 3.$ If all dihedral angles of $\Omega_n$ are submultiples of $\pi,$ then there exists a parallelopiped $\tilde \Omega_n$ generated by $n$ linearily independent vectors $\{\mu_j\}_{j=1}^{n}$ in $R^n$ containing $\Omega_n$ so that solutions of $\Delta u+ \la u=0$ in $\Omega_n$ with either the boundary condition $u=0$ or $\partial u/\partial n=0$ are expressed by linear combinations of those of $\Delta u+ \la u=0$ in $\tilde \Omega_n$ with periodic boundary condition. Moreover, if $\{\mu_j\}_{j=1}^{n}$ satisfies rational condition, we guarantee the weaker PSA for the domain $\Omega_n.$

Keywords : discrete group, inertial manifold, fundamental polyhedron, the weaker principle of spatial averaging